Past Industrial and Applied Mathematics Seminar

The industrial prilling process is amongst the most favourite technique employed in generating monodisperse droplets. In such a process long curved jets are generated from a rotating drum which in turn breakup and from droplets. In this talk we describe the experimental set-up and the theory to model this process. We will consider the effects of changing the rheology of the fluid as well as the addition of surface agents to modify breakup characterstics. Both temporal and spatial instability will be considered as well as nonlinear numerical simulations with comparisons between experiments.

The inverse acoustic obstacle scattering problem, in its most general

form, seeks to determine the nature of an unknown scatterer from knowl-

edge of its far eld or radiation pattern. The problem which is the main

concern here is:

If the scattering cross section, i.e the absolute value of the radiation

pattern, of an unknown scatterer is known determine its shape.

In this talk we explore the problem from a number of points of view.

These include questions of uniqueness, methods of solution including it-

erative methods, the Minkowski problem and level set methods. We con-

clude by looking at the problem of acoustically invisible gateways and its

connections with cloaking

A central challenge in socio-physics is understanding how groups of self-interested agents make collective decisions. For humans many insights in the underlying opinion formation process have been gained from network models, which represent agents as nodes and social contacts as links. Over the past decade these models have been expanded

to include the feedback of the opinions held by agents on the structure of the network. While a verification of these adaptive models in humans is still difficult, evidence is now starting to appear in opinion formation experiments with animals, where the choice that is being made concerns the direction of movement. In this talk I show how analytical insights can be gained from adaptive networks models and how predictions from these models can be verified in experiments with swarming animals. The results of this work point to a similarity between swarming and human opinion formation and reveal insights in the dynamics of the opinion formation process. In particular I show that in a population that is under control of a strongly opinionated minority a democratic consensus can be restored by the addition of

uninformed individuals.

Brittle failure through multiple cracks occurs in a wide variety of contexts, from microscopic failures in dental enamel and cleaved silicon to geological faults and planetary ice crusts. In each of these situations, with complicated stress geometries and different microscopic mechanisms, pairwise interactions between approaching cracks nonetheless produce characteristically curved fracture paths. We investigate the origins of this widely observed "en passant" crack pattern by fracturing a rectangular slab which is notched on each long side and then subjected to quasistatic uniaxial strain from the short side. The two cracks propagate along approximately straight paths until they pass each other, after which they curve and release a lens-shaped fragment. We find that, for materials with diverse mechanical properties, each curve has an approximately square-root shape, and that the length of each fragment is twice its width. We are able to explain the origins of this universal shape with a simple geometrical model.

This work builds on the foundation laid by Benney & Timson (1980), who

examined the flow near a contact line and showed that, if the contact

angle is 180 degrees, the usual contact-line singularity does not arise.

Their local analysis, however, does not allow one to determine the

velocity of the contact line and their expression for the shape of the

free boundary involves undetermined constants - for which they have been

severely criticised by Ngan & Dussan V. (1984). As a result, the ideas

of Benny & Timson (1980) have been largely forgotten.

The present work shows that the criticism of Ngan & Dussan V. (1984)

was, in fact, unjust. We consider a two-dimensional steady Couette flow

with a free boundary, for which the local analysis of Benney & Timson

(1980) can be complemented by an analysis of the global flow (provided

the slope of the free boundary is small, so the lubrication

approximation can be used). We show that the undetermined constants in

the solution of Benney & Timson (1980) can all be fixed by matching

their local solution to the global one. The latter also determines the

contact line's velocity, which we compute among other characteristics of

the global flow.

Understanding the fatigue of metals under cyclic loads is crucial for some fields in mechanical engineering, such as the design of wheels of high speed trains and aero-plane engines. Experimentally it has been found that metal fatigue induced by cyclic loads is closely related to a ladder shape pattern of dislocations known as a persistent slip band (PSB). In this talk, a quantitative description for the formation of PSBs is proposed from two angles: 1. the motion of a single dislocation analised by using asymptotic expansions and numerical simulations; 2. the collective behaviour of a large number of dislocations analised by using a method of multiple scales.

It is an inherent premise in Boltzmann's formulation of linear viscoelasticity, that for shear deformations at constant pressure and constant temperature, every material has a unique continuous relaxation spectrum. This spectrum defines the memory kernel of the material. Only a few models for representing the continuous spectrum have been proposed, and these are entirely empirical in nature.

Extensive laboratory time is spent worldwide in collecting dynamic data from which the relaxation spectra of different materials may be inferred. In general the process involves the solution of one or more exponentially ill-posed inverse problems.

In this talk I shall present rigorous models for the continuous relaxation spectrum. These arise naturally from the theory of continuous wavelet transforms. In solving the inverse problem I shall discuss the role of sparsity as one means of regularization, but there is also a secondary regularization parameter which is linked, as always, to resolution. The topic of model-induced super-resolution is discussed, and I shall give numerical results for both synthetic and real experimental data.

The talk is based on joint work with Neil Goulding (Cardiff University).

Tsunami asymptotics: For most of their propagation, tsunamis are linear dispersive waves whose speed is limited by the depth of the ocean and which can be regarded as diffraction-decorated caustics in spacetime. For constant depth, uniform asymptotics gives a very accurate compact description of the tsunami profile generated by an arbitrary initial disturbance. Variations in depth can focus tsunamis onto cusped caustics, and this 'singularity on a singularity' constitutes an unusual diffraction problem, whose solution indicates that focusing can amplify the tsunami energy by an order of magnitude.

Initial stage of the flow with a free surface generated by a vertical

wall moving from a liquid of finite depth in a gravitational field is

studied. The liquid is inviscid and incompressible, and its flow is

irrotational. Initially the liquid is at rest. The wall starts to move

from the liquid with a constant acceleration.

It is shown that, if the acceleration of the plate is small, then the

liquid free surface separates from the wall only along an

exponentially small interval. The interval on the wall, along which

the free surface instantly separates for moderate acceleration of the

wall, is determined by using the condition that the displacements of

liquid particles are finite. During the initial stage the original

problem of hydrodynamics is reduced to a mixed boundary-value problem

with respect to the velocity field with unknown in advance position of

the separation point. The solution of this

problem is derived in terms of complete elliptic integrals. The

initial shape of the separated free surface is calculated and compared

with that predicted by the small-time solution of the dam break

problem. It is shown that the free surface at the separation point is

orthogonal to the moving plate.

Initial acceleration of a dam, which is suddenly released, is calculated.

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The mechanisms for the selection of the propagation speed of waves

connecting unstable to stable states will be discussed in the

spatially non-homogeneous case, the differences from the very

well-studied homogeneous version being emphasised.